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337. 1. Thomas has 20 books and John 5. Thomas has how many times as many as John?

2. Peter is 16 years old and his brother 8. How do their ages compare?

3. What part of 27 feet is 9 feet ?

4. How does 20 compare with 5? 24 with 6? 27 with 9? 5. What is the relation of $35 and $7? Of 51 miles and 17 miles? Of 7 pounds and 42 pounds?

338. Ratio is the relation of two like numbers shown by their quotient.

It is determined by dividing the first by the second. Thus,

The ratio of 15 to 5 is 15 ÷ 5, or 3.

339. Ratio is usually indicated by :, which is an abbreviated form of Thus,

186 expresses the ratio of 18 to 6.

340. The Terms of a ratio are the two numbers compared. The Antecedent is the first term of a ratio, the Consequent is the second term, and the two terms together are called a Couplet.

341. An Inverse Ratio is a ratio formed by inverting the terms of a given ratio. Thus,

89 is the inverse of 9: 8.

342. A Simple Ratio is the ratio of two numbers. Thus, 21:37 is a simple ratio.

343. A Compound Ratio is the product of two or more simple ratios. It is usually indicated by means of the brace. Thus,

8:2)

:

9:3 or 8 × 9 × 10 2 x 3 x 5 is a compound ratio, 10:5

equal to the simple ratio 720 : 30.

A compound ratio may be changed to a simple ratio by multiplying antecedents together for a new antecedent, and consequents for a new consequent.

A ratio, like a fraction, is simply an indicated division (Art. 108). The principles of common fractions are equally applicable to ratios, the antecedent being the numerator and the consequent the denominator (Arts. 111, 114).

344. Principles of Ratio.

1. The ratio is equal to the antecedent divided by the consequent.

2. The consequent is equal to the antecedent divided by the ratio.

3. The antecedent is equal to the consequent multiplied by

the ratio.

4. Multiplying or dividing both the antecedent and the consequent by the same number does not change the ratio.

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15. What is the inverse ratio of 65: 15?

16. What is the inverse ratio of 25: 625?

17. Which is the greater ratio, 12 to 13 or 25 to 27 ?

18. If 6.25 is the antecedent and 5 the ratio, what is the consequent ?

19. If 27 is the consequent and the ratio, what is the antecedent?

20. What is the value of the compound ratio

4:10

8: 6 } ? 21. What is the ratio compounded of (68) × (16:10) × (12:9) ?

PROPORTION.

22. The ratio of 21 to 7 is what number? Of 51 to 17? 23. Name two numbers having the same ratio as 51 to 17. 24. What number has the same relation to 17 as 21 has to 7 ?

25. If 12 yards of cloth cost $40, what part of $40 will 3 yards cost?

26. How does the ratio of 3 yards to 12 yards compare with the ratio of $10 to $40?

345. A Proportion is an equality of ratios. Thus,

=

123 40 10 is a proportion.

The equality of ratios may be indicated either by or ::. Thus,

8:2 16:4, or 8 : 2 :: 16: 4.

means 8 to 2 equals 16 to 4, or 8 is to 2 as 16 is to 4.

=

346. Each term of a proportion is called a Proportional; the first and fourth terms are called Extremes; and the second and third terms, Means.

When the two means are the same number, that number is a Mean Proportional between the two extremes. Thus, In 12 6 6 3, 6 is a mean proportional between 12 and 3.

:

= :

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As these fractions are equal and their denominators alike, their numerators must be equal, or 6 x 2 = 3 × 4. X But 6 and 2 are the extremes, and 3 and 4 the means. Hence the following

348. Principles of Proportion.

1. In a proportion the product of the means is equal to the product of the extremes.

2. Either extreme is equal to the product of the means divided by the other extreme.

3. Either mean is equal to the product of the extremes divided by the other mean.

WRITTEN EXERCISES.

Find the missing term represented by x in the following proportions:

27. 14:7 18 : x.
28. 5:20 =x: 60.
29. x: 865: 13.
30. 648 243 = 24: x.

31. x 4: 8.

32. $45 $24 : 15 yd. : x.

:

33. x $9: 60 men : 18 men. 34. 5 tonston :: x: $7.50.

SIMPLE PROPORTION.

349. A Simple Proportion is an equality between two simple ratios.

It applies to the solution of questions in which three terms of a proportion are given to find the fourth.

NOTE. - Of the given terms two must be of the same kind, and constitute a ratio; and the other must be of the same kind as the required term, and constitute with it another ratio equal to the first.

WRITTEN EXERCISES.

35. If 37 yards of cloth cost $ 111, what will 19 yards cost?

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Solution. As 37 yards must evidently have the same ratio to 19 yards that $111, the cost of 37 yards, has to the cost of 19 yards, or the answer, we arrange the terms so as

to express the equality of these ratios. Or,

As the fourth term is to be dollars, we make $111 the third term. The fourth term is to be less than the third term, because 19 yards will cost less than 37 yards. Hence the second term must be smaller than the first, and the first ratio is 37: 19. Dividing the product of the means by the given extreme, we have as the answer $57. Or,

If 37 yards cost $111, 1 yard costs of $111, and 19 yards 19 times as much, or 3 of $ 111, or $57.

36. If 12 barrels of apples cost $51, what will 30 barrels cost?

37. If the rent of 183 acres of land is $ 273, what will be the rent of 61 acres?

38. What number of men will be required to perform in 16 days a piece of work that would take 30 men 48 days?

39. If 24 men can mow a field in 15 days, how many days will it take 20 men to do it?

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